We shall describe how to construct a fundamental solution for the Pell equation $x^2-m{y}^2=1$ over finite fields of characteristic $p\ne 2$. Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation $x^2-m{y}^2=n$.